On the use of incomplete semiiterative methods for singular systems and applications in Markov chain modeling Yimin Wei a, * , Hebing Wu b a Department of Mathematics, Fudan University, Shanghai 200433, People's Republic of China b Institute of Mathematics, Fudan University, Shanghai 200433, People's Republic of China Abstract There are several methods for solving linear ®xed point problem x  Tx  c, where T 2 C N;N is a square matrix and I  T is possibly singular. Such problems arise if one splitsthecoecientmatrixofasingularsystem Ax  b of algebraic equations according to A  M  N M nonsingular) which leads to x  M 1 Nx  M 1 b : Tx  c. The basic iteration x 0 2 C N , x m  Tx m1  c m P 1 requires the modulus of every eigenvalue of the iteration matrix T except1islessthan1and q  indexI  T ,theindexof I  T is equal to 1 for convergence. In this paper, we try to use the incomplete semiiterative methods ISIM) to solve x  Tx  c when c 2 RI  T  q . Usually the special semiiter- ative methods are convergent even when the spectral radius of the iteration matrix is greater than 1 and q P 1. Then the use of the ISIM in the Markov chain modeling is considered. Finally, numerical examples are reported. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Singular systems; Index; Drazin inverse; Semiiterative method; Incomplete semiiterative method; Markov chain Applied Mathematics and Computation 125 2002) 245±259 www.elsevier.com/locate/amc * Corresponding author. E-mail addresses: ymwei@fudan.edu.cn Y. Wei), 960167@fudan.edu.cn H. Wu). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0096-300300)00127-2